The Guess The Number Puzzle

Are you a fan of puzzles? In early 2014 over 1,600 of Pinnacle’s customers said yes and participated in a game theory puzzle that invited them to pick a whole number from 0 to 100 with the aim of getting as close as possible to two-thirds of the average of all guesses. The results revealed a lot about how people struggle with logic and accounting for other people's behaviour; a great analogy for betting markets.

Guess the Number sounds like a simple proposition. It is, however, a challenging mental computation that requires both deductive reasoning and the ability to predict the behaviour of other participants.

These skills are extremely valuable in betting, where you need to make your own rationality assessment of the odds as well as accounting for the influence of other bettors' - whose approach may not be as rational as yours.

The Premise

The premise of Guess the Number is simple:

We asked participants to guess a whole integer from 0-100 inclusive that is closest to two-thirds of the average of all guesses.

The simplicity can often be the biggest source of confusion, which is evident in the number of implausible answers.

The Game Theory

Considering that 100 was the maximum number in the available range that could be selected, it was mathematically impossible for the final number to be anything above 66 (Round numbers only were accepted for the purposes of this puzzle) because the aim was to get to 2/3 of the average of other guesses.

A certain number of participants failed to understand that point, and skewed the outcome. This is often what happens in betting, when uninformed “square” bettors can exert influence on the odds.

If we assume that the participants would pick a random number between 0-100, that would give an average of 50, making the correct answer of the puzzle 33.

Since the number, however, was not supposed to be randomly picked, those who incorporated a second level of processing in their thinking, would have assumed that the winning number should be lower than that.

Theoretically speaking, if all participants were purely rational and expected the rest of the players to be rational too, the winning answer would be 0.

The beauty of the puzzle, though, is that not everybody is rational and this has to be priced in the guessing process in order to stand a chance to win.

Reality check

As shown in the chart below, the most popular answer was 33, which corresponds to first level thinking.

There was also a 2% of guesses that fell between 67-100, which is mathematically impossible, therefore representing the players who didn't comprehend the game.

2.5% of the participants went for the mathematically sophisticated answer of 0, which however did not incorporate the number of irrational players.

With the correct answer being 20, the winners proved to be those who both understood the game and also priced in the behavior of the participants who didn't.

Wisdom of the crowd

"Guess the Number" is a competition with a long history. The 1997 Financial Times version of it produced a winning number of 13, suggesting a more mathematically sophisticated audience.

One could argue, however, that a first prize worth over $10,000 would inevitably require more attention to the workings of the game. Another version run in Denmark for a prize of 5,000 Danish kroner produced a winning number of 21.6.

Guess the Number has a rich history, and our version of the puzzle - to an audience of bettors - caught the attention of Richard Thaler, one of the Godfather's of behavioural science, as you can see from his recent tweet.

What does all that mean for betting? As with the Guess the Number quiz, the results of sporting events are unknown, with a large crowd of bettors trying to provide an assessment of the respective probabilities of each outcome, leading to odds fluctuations.

The bettors who cause the markets to move come from a variety of knowledge and experience backgrounds, with players both over- and under-estimating outcomes.

Although, it is reasonable to assume that the more bettors participate in a market, the more likely the odds are to represent the true probability of the outcome of an event, what happens when crowds act like herds?

Guess the Number illustrated that when faced with logic based decisions, people act irrationally, but it is equally irrational to overlook this in your own assessment; being smart isn’t enough you need to be smart enough to understand and account for other people’s shortcomings, and this is crucial in betting.

We have a team of editors and writers at Pinnacle, as well as a collection of external contributors, ranging from university lecturers and renowned authors, to ex-traders and esteemed sports experts. Collectively, the Pinnacle team and external contributors produce the educational content within Betting Resources.

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